What is the critical value of z for a 99% confidence interval for a known standard deviation? B) A 99% confidence interval for the mean of a population was found to be (123.33, 140.67). If the sample size was 40, find the margin of error for this estimation. C) A 95% confidence interval for a population mean is (12.9, 18.6). Which of the following is the sample mean used to calculate this interval?
The Correct Answer and Explanation is:
Critical Value of z for a 99% Confidence Interval
To find the critical value of z for a 99% confidence interval, we need to consider that a 99% confidence interval means that 99% of the data lies within the interval, and the remaining 1% is split equally on both sides of the distribution. Therefore, there is 0.5% of the data in each tail of the standard normal distribution.
For a 99% confidence interval, we look up the z-value corresponding to a cumulative probability of 0.995 (because 0.5% is in the right tail, so 99.5% is to the left). From the standard normal distribution table or using a calculator, the z-value is approximately 2.576. This means the critical value of z for a 99% confidence interval is 2.576.
B) Margin of Error for the Confidence Interval
The margin of error (ME) is the half-width of the confidence interval. It can be calculated using the formula:Margin of Error=Upper Bound−Lower Bound2\text{Margin of Error} = \frac{\text{Upper Bound} – \text{Lower Bound}}{2}Margin of Error=2Upper Bound−Lower Bound
For the given confidence interval of (123.33, 140.67), the margin of error is:ME=140.67−123.332=17.342=8.67\text{ME} = \frac{140.67 – 123.33}{2} = \frac{17.34}{2} = 8.67ME=2140.67−123.33=217.34=8.67
Therefore, the margin of error for this estimation is 8.67.
C) Sample Mean from the Confidence Interval
The sample mean (xˉ\bar{x}xˉ) is the midpoint of the confidence interval. To find the sample mean, we average the upper and lower bounds of the confidence interval:xˉ=Upper Bound+Lower Bound2\bar{x} = \frac{\text{Upper Bound} + \text{Lower Bound}}{2}xˉ=2Upper Bound+Lower Bound
For the given confidence interval of (12.9, 18.6):xˉ=18.6+12.92=31.52=15.75\bar{x} = \frac{18.6 + 12.9}{2} = \frac{31.5}{2} = 15.75xˉ=218.6+12.9=231.5=15.75
Thus, the sample mean used to calculate the 95% confidence interval is 15.75.
Summary of Answers:
- A) The critical value of z for a 99% confidence interval is 2.576.
- B) The margin of error for the given confidence interval is 8.67.
- C) The sample mean for the 95% confidence interval is 15.75.
